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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1962 Kurschak Competition
1962 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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1of DA, DB, DC exceeds 1 of PA, PB and PC, tetrahedron ABCD
P
P
P
is any point of the tetrahedron
A
B
C
D
ABCD
A
BC
D
except
D
D
D
. Show that at least one of the three distances
D
A
DA
D
A
,
D
B
DB
D
B
,
D
C
DC
D
C
exceeds at least one of the distances
P
A
PA
P
A
,
P
B
PB
PB
and
P
C
PC
PC
.
2
1
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n+1 diagonals of a convex n-gon
Show that given any
n
+
1
n+1
n
+
1
diagonals of a convex
n
n
n
-gon, one can always find two which have no common point.
1
1
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no of positive divisors of n^2 = no of ordered pairs (a, b) with lowest lcm n
Show that the number of ordered pairs
(
a
,
b
)
(a, b)
(
a
,
b
)
of positive integers with lowest common multiple
n
n
n
is the same as the number of positive divisors of
n
2
n^2
n
2
.